Vertex-Compressed Subalgebras of a Graph von Neumann Algebra

Abstract

In Cho (Acta Appl. Math. 95:95–134, 2007 and Complex Anal. Oper. Theory 1:367–398, 2007), we introduced Graph von Neumann Algebras which are the (groupoid) crossed product algebras of von Neumann algebras and graph groupoids via graph-representations, which are groupoid actions. In Cho (Acta Appl. Math. 95:95–134, 2007), we showed that such crossed product algebras have the amalgamated reduced free probabilistic properties, where the reduction is totally depending on given directed graphs. Moreover, in Cho (Complex Anal. Oper. Theory 1:367–398, 2007), we characterize each amalgamated free blocks of graph von Neumann algebras: we showed that they are characterized by the well-known von Neumann algebras: Classical group crossed product algebras and (operator-valued) matricial algebras. This shows that we can provide a nicer way to investigate such groupoid crossed product algebras, since we only need to concentrate on studying graph groupoids and characterized algebras. How about the compressed subalgebras of them? i.e., how about the inner (cornered) structures of a graph von Neumann algebra? In this paper, we will provides the answer of this question. Consequently, we show that vertex-compressed subalgebras of a graph von Neumann algebra are characterized by other graph von Neumann algebras. This gives the full characterization of the vertex-compressed subalgebras of a graph von Neumann algebra, by other graph von Neumann algebras.